Variance

We can use variance to find out the degree of deviation of values from the mean value. It is represented by the symbol σ2 and calculated by squaring the standard deviation. Low variance indicates the numbers are close to one another, and high variance means the numbers are dispersed. Some key takeaways of variance are listed below: 

• Variance measures the difference between numbers in a data set.
   
• It specifically quantifies how data spreads around the sample mean. 
   
• Investors use variance to assess investment risks and potential profits. 
   
• It is also used to analyze the relative performance of each asset in a portfolio and determine optimal asset allocation. 
   
• The square root of variance is determined to find the standard deviation. 

Variance is used to measure how much of the data actually varies from each other. It can be calculated by finding the average of the squared differences from the mean. We have to follow the below mentioned steps to find the variance of a set of values. 

Step 1: We have to find the mean in the first step. The mean can be calculated by dividing the number of values by the sum of all values. 
          Mean = Sum of all values / Total number of values

Step 2: Determine the squared deviations of the data values from the mean. To get the deviation, subtract the mean from each score. 
        (Data value - Mean)2

Step 3: Determine the data set’s variance, which is the mean of the squared deviations from the given values. We can find the square by multiplying each deviation by itself from the mean. As a result, we will get positive numbers. 

Step 4: Calculate the sum of squares by adding up all the squared deviations. 

Step 5: To calculate a sample variance, divide the sum of the squared deviations by (n -1). To determine a population variance, divide the sum by N.

Now, let us examine the formulas used to find the variance. 
The formula for calculating the population variance is: 

(σ2) = ∑(xi - μ)2/ N

Here, σ2 = Population variance 
xi = Each individual value
μ = Population mean
N = Total number of values in the population
Next, the formula for sample variance is:

s2 =  (xi - x̄)2/ n -1
Here, s2 = Sample variance
xi = Each individual value
x̄ = Sample mean
n = Total number of values in the sample
n - 1 = Degrees of freedom

In reference to the mean, variance plays an important role in measuring the spread of data points. It helps us in understanding the consistency and variability of data by assessing the deviations from the mean. On the basis of data set, variance can be of two types: Population  (σ2) and sample variance (s2). 

Population variance (σ2) 

It calculates the population’s overall dispersion. Population variance determines how the data points are distributed in the population. A population symbolizes a group of individuals. This variance shows how the group’s population is different from mean population. 
Each data point’s squared distance from the population mean is calculated by the population variance. The formula for population variance is: 
Population variance (σ2) =  (xi - μ)2/ N

Sample variance

Calculating population variance becomes challenging when the population data is too large. Instead, we use sample variance, which refers to a sample from the dataset and we calculate its variance. While doing this, rather than the population mean we use the sample mean. Sample variance is the average of the squared differences between the sample data points and the sample mean. The formula for sample variance is: 
s2 =  (xi - x̄)2/ n -1

Variance measures variability by averaging the squared deviations from the mean. Population variance and sample variance are the two types of variance. The main differences between these two types are given below:

• Population variance calculates the variance for the whole population. Whereas, a sample variance uses a sample or subset of the population to measure the variance. 
   
• For population variance, an accurate figure is calculated if the data is gathered from every member of the population. 
• The formula for population variance is (σ2) =  (xi - μ)2/ N and the formula for sample variance is s2 =  (xi - x̄)2/ n -1. 
   
• In the population variance, the total number of data points (N) is divided by the value. While in the sample variance, the value is divided by N -1. 
   
• The sample variance is usually less than the actual variance of the population. The true variance of the total population is higher than the variance calculated from a sample.
   
• Population variance is used in census studies and data analysis when population-wide data is available.
   
•  Sample variance is used in research sampling, experiments, and surveys when estimating population variance through sample analysis. 
   

To evaluate the data and make well-informed decisions, variance is used in various fields. The real-world importance of variance is countless. They are listed as follows:

•  In the field of investment and finance, variance is a tool to assess stock risk. A stock with high variance indicates high risk and a low variance indicates lower risk. 
   
• To verify the consistency of the products, the manufacturers use the variance. If there is a high variance, it indicates the production process needs to be modified.
   
• The teachers and the educational institutions can employ the variance to assess the performance of their students. If there is a low variance, it indicates that there is a small difference between the top and low performers.
   
• Weather forecasters and sports professionals use variance to analyze the temperature changes over time and evaluate the consistency and performance of players.